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Group theoretical aspects of the geometric phase in quantum mechanics.

Mukunda, N (1999) Group theoretical aspects of the geometric phase in quantum mechanics. In: Proceedings of the XXII International Colloquium in Group Theoretical Methods in Physics, 1998, Hobart, pp. 86-96.

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Abstract

The geometric phase in quantum mechanics was originally elucidated in the context of the adiabatic theorem. This and other limitations were later removed. We show how this phase can be presented as the simplest functional of smooth open curves of unit vectors in Hilbert space, invariant under the two groups of local phase changes and monotonic reparametrisations. This viewpoint clarifies the role of the early Pancharatnam criterion for two Hilbert space vectors to be `in phase', in the later geometric phase context. Noncyclic evolution is handled in a very direct manner. Connections to the Bargmann invariants of quantum mechanics are immediate and can be exploited. A class of geometric phases arises in connection with unitary representations of Lie groups. In this case a surprisingly complete analysis of the structure of the geometric phase is possible, motivated by the Wigner-Eckart theorem. The interplay between geometric phase ideas and Lie group unitary representations is extraordinarily rich; the differential geometric structures of Lie groups and coset spaces play very important roles and allow classification of all possible geometric phases associated with a given Lie group. Examples involving SU(2), SU(3) and the metaplectic group Mp(2) are presented.

Item Type: Conference Paper
Department/Centre: Division of Physical & Mathematical Sciences > Centre for Theoretical Studies
Date Deposited: 12 Jul 2004
Last Modified: 10 May 2011 07:16
URI: http://eprints.iisc.ernet.in/id/eprint/746

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